List of 7-prime limit accidentals

List of 7-prime limit accidentals

Following are two lists with ratios up to the 7-prime limit, and corresponding accidentals, sorted by odd limit:

The first list contains ratios with an accurate representation in the Olympian set

The second list contains other important or potentially useful ratios. Since there are no corresponding accidentals in the Olympian set, the list contains slightly inaccurate accidentals that usually represent different ratios.

When the Standard Sagittal notation with its seven Pythagorean nominals is used, a much smaller set of accidentals is usually sufficient to notate microtonal music. However, in linear temperament notations, accidentals usually represent a chroma or stacked chromas for various MOS or MODMOS scales. That means unusual accidentals may be required, so it's good to have an extensive list of accidentals available.

When working with temperament notations, it's important to use accidentals that work with any possible tuning for the temperament. So in that case, it is strongly recommended to use accurate accidentals from the first list. The inaccurate ratios from the second list should only be considered if it is clear which ratio they are refering to.

This post is work-in-progress, and currently incomplete. Here is my todo list:

Add systematic comma names

Maybe extend the second list

7-limit ratios with an accurate representations in the Olympian symbol set: (currently incomplete)

I have edited this post a lot, so here's the text from the previous version, for reference:

List of 7-prime limit / 6125-odd limit accidentals

When working a lot with MOS notations, I find it vital to have a list of simple 7-limit accidentals. So I started to compile a list, and decided to post it here so I can share it with others, or ask for help to complete / improve it.

The ratios in the list are 7-prime limit, 6125-odd limit, and not larger than a 9/8 major tone. This list isn't complete, and only contains ratios that I find either important, or potentially useful. I list accidental symbols / characters where I could find them.

Note: This list isn't intended as a reference for standard Sagittal notation. A lot of the accidentals in this list aren't required in standard Sagittal notation, and using more different types of accidentals than necessary can make it difficult for others to read a score.

On the contrary, MOS notations (or other non-standard notations) sometimes require unusual accidentals, and can become unwieldy if multiple accidentals have to be used for a single chroma, or if stacks of chromas are common, and can't be reduced to fewer accidentals (for example, there are considerable chroma stacks when notating 87-edo with a Rodan[5] notation).

Edit note: I started to compile a list of 7-limit ratios with representations in the Olympian set, and moved ratios that are not contained in this set into a second list. This is work in progress, so the lists are currently incomplete!

Edit: The following points have already been solved or addressed:

One accidental pair which I'd be interested in is for 256:245, the difference between 8/7 and 35/32. I'm working a lot with scales where the major third 5/4 is divided into 35/32 and 8/7. Examples where 256:245 accidentals can become useful are:

Consider a notation based on the 6-note MOS 1/1 28/25 5/4 7/5 25/16 7/4 for the 2.5.7 temperament that tempers out the hemimean comma 3136/3125 (i.e. 5/4 is split into two 28/25 steps). Two chromas could be expressed as 256/245. This notation could be used as a basis for a hemithirds, hemiwürschmidt, or roulette notation. A roulette[6] notation could be used as a 2.5.7.11.13 notation (no-fifths 13-limit) for 37-edo, without having to decide for a fifth (i.e. between 37p and 37b), and it shares the same mapping with 74- and 111-edo.

I'm currently experimenting with a 7-limit JI notation based on the following pair of max-variety-4 scales: 1/1 35/32 5/4 21/16 3/2 105/64 15/8, and 1/1 35/32 5/4 21/16 3/2 105/64 7/4. Both scales are mirror symmetric, and can be obtained by removing a pitch from 1/1 35/32 5/4 21/16 3/2 105/64 7/4 15/8, which is the octave-reduced version of the complete chord 1:3:5:7:15:21:35:105. Having a single accidental for 256/245 would be extremely useful for this notation. I'm planning to use it as a basis for several temperaments that are supported by 31-edo, including the planar temperaments hewuermity, gamelan and marvel, and the linear temperaments mohajira, orwell, valentine, miracle and mothra, probably more. If it works out, this notation should help me considerably with 31-based compositions with modulations between scales of different temperaments.

For Moments of Symmetry based on pentatonic scales, it would be very useful to have accidentals for 16:15 and 256:243.

I think I'd prefer to use two accidentals for neutral or major second-sized ratios like 160/147 or 9/8, as an indicator for large (or uncommon) alterations. For example, 160/147 could be expressed as 15/14 * 64/63, or 25/24 * 256/245.

A question: Herman Miller mentioned accidentals for 21:20 that I couldn't find anywhere else. Is there a list of Sagittal accidentals that I don't know about?

P.S.: Apparently, some Sagittal accidentals aren't implemented as emoticons yet, or don't display for other reasons.

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Wed Nov 02, 2016 5:21 pm

by Dave Keenan

That's marvelous work, Xen.

Xen-Gedankenwelt wrote:For Moments of Symmetry based on pentatonic scales, it would be very useful to have accidentals for 16:15 and 256:243.

15:16 is
256:243, the limma, is

I think I'd prefer to use two accidentals for neutral or major second-sized ratios like 160/147 or 9/8, as an indicator for large (or uncommon) alterations. For example, 160/147 could be expressed as 15/14 * 64/63, or 25/24 * 256/245.

8:9 is an apotome plus a limma so that's + =
147:160 is + =

A question: Herman Miller mentioned accidentals for 21:20 that I couldn't find anywhere else. Is there a list of Sagittal accidentals that I don't know about?

20:21 is greater than 68.57 cents (half an apotome plus half a pythagorean comma) so it will not have a single shaft symbol. It is not greater than an apotome 2^11 : 3^7 so it will have a double-shaft symbol. This double shaft symbol will be the symbol for the apotome-complement of 3^7/2^11 * 20/21 = 3645/3584 which we call the 5:7 comma. You can see here at around 29 cents, that its symbol is . And in figure 13 of the XH article you can see that the complement of is and so the complement of is and therefore the symbol for 20:21 is as you have it.

Can you give a link to Herman Miller's supposed alternatives? I suspect you may be misunderstanding something he wrote.

P.S.: Apparently, some Sagittal accidentals aren't implemented as emoticons yet, or don't display for other reasons.

Those whose long ASCII includes two consecutive slashes // or sloshes \\ will display correctly if you put a space between the slashes or sloshes. This is necessary because of some other forum features, including automatic URL parsing, that treat double-slash and double-slosh as special cases.

I'm sorry that 3-shaft and X-shaft Sagittals are not yet implemented as emoticons, except for the 72-edo set. You can see this if you click View more smilies, which appears on the right when you are editing a post.

I'd be very grateful if you would help me out with that. The attached .png file contains the required bitmapped upward symbols at the required size. It just needs someone to separate the remaining 3-shaft and X-shaft symbols into individual .png files with appropriate names, and copy and flip them to create the downward symbols.

If you move your cursor over an existing symbol in the "View more smilies" palette, its Sagispeak name will pop up. This name is also the name of its .png file, with any spaces replaced by underscores. In cases where there are alternative names, it is the first and longest name that is used for the .png file name. You should be able to work out the required names for the new .png files from this and the Sagispeak documentation.

Each .png file must be 19 pixels high, with upward symbols set as low as possible and downward symbols set as high as possible. There should be one pixel of whitespace to the left and none to the right. The background for the symbol needs to be transparent, not white. I have used the free Paint.NET for this in the past.

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Wed Nov 02, 2016 7:24 pm

by Dave Keenan

Xen-Gedankenwelt wrote:One accidental pair which I'd be interested in is for 256:245, the difference between 8/7 and 35/32.

245:256 is which is the apotome-complement of i.e. =
9:10 is =

You should be able to get the rest from George's spreadsheet, using figure 13 of the XH paper to convert from mixed to pure, as I have done above.

A question: Herman Miller mentioned accidentals for 21:20 that I couldn't find anywhere else. Is there a list of Sagittal accidentals that I don't know about?

I think I found what you are referring to, here: viewtopic.php?p=477&hilit=21/20#p477. would be a valid symbol for 20:21 at the Promethean level (the highest precision level that does not use accent marks). George's spreadsheet gives Promethean which is equivalent. But I agree we should stick to the Olympian level for this list.

May I suggest you eliminate the comma punctuation marks from your "accidentals" column in future. I think they look too much like accent marks. At least, they confuse my eyes.

It would be good to indicate in some way when the given accidentals are not an exact representation of the given comma. This is the case for the symbol I gave for 147:160, the 5:49-comma-plus-apotome, . That symbol's primary role is 0.12 cents larger, the 7:11-comma-plus-apotome 92274688:100442349. I'm pretty sure that's the only one so far.

Yes, I used this list, along with the Sagittal2_character_map.pdf file from the Sagittal website. I'm don't know much about how the symbols are constructed, but I think (or at least hope) I'll figure that out from the links and instructions you gave me.

So the list is somewhat arbitrary, without using a particular algorithm. That means not all ratios are necessarily important, and some important ratios may be missing.

I should be able to do the image conversions, using MS Paint for aligning and mirroring, and using Gimp to add transparency. I'll try to figure out the proper names, but if I fail I can still name the files something like "3-shaft-7-down.png" for the down-version of the 7th accidental from the left in the row of 3-shaft accidentals.

Dave Keenan wrote:

A question: Herman Miller mentioned accidentals for 21:20 that I couldn't find anywhere else. Is there a list of Sagittal accidentals that I don't know about?

I think I found what you are referring to, here: viewtopic.php?p=477&hilit=21/20#p477. would be a valid symbol for 20:21 at the Promethean level (the highest precision level that does not use accent marks). George's spreadsheet gives Promethean which is equivalent. But I agree we should stick to the Olympian level for this list.

Yes, that's what I was refering to. He also mentioned the accidentals in the 37-edo thread. I noticed that one accidental type is already used for another comma, so I added the other one to my list - I'm glad to learn that this one is not ambigous.

Dave Keenan wrote:May I suggest you eliminate the comma punctuation marks from your "accidentals" column in future. I think they look too much like accent marks. At least, they confuse my eyes.

Done. I wasn't sure which other separator to use, but finally decided to omit it for the symbols, and use "&" between multi-character accidentals.

Dave Keenan wrote:It would be good to indicate in some way when the given accidentals are not an exact representation of the given comma. This is the case for the symbol I gave for 147:160, the 5:49-comma-plus-apotome, (||| . That symbol's primary role is 0.12 cents larger, the 7:11-comma-plus-apotome 92274688:100442349. I'm pretty sure that's the only one so far.

I agree this is important here, thanks for the information!

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Thu Nov 03, 2016 7:06 pm

by Dave Keenan

I think your asterisk note above should say "* ambiguous, usually refers to a different comma" rather than "... different accidental".

So the list is somewhat arbitrary, without using a particular algorithm. That means not all ratios are necessarily important, and some important ratios may be missing.

I note that the commas listed in the wiki tend to be chosen because they vanish in some temperament, and they tend to be named after the temperament in which they vanish. Of course, when looking for commas for use in MOS notations, we need commas that do not vanish. The properties that make commas useful for notation are in some senses the opposite of those that make them useful for creating temperaments. And we would like the names of the commas to convey information about what they notate, rather than where they vanish.

I suggest a slightly different approach to this table, which would also eliminate any concerns about ambiguous symbols. You could simply list all 7-limit commas that have an exact representation in Olympian Sagittal. Begin with the list of Olympian ratios for single-shaft symbols in either sag_ji4.par or George's Sagittal-JI.xls spreadsheet, and delete all the rows that are not 7-limit. Then add rows for all the double-shaft symbols which are apotome-complements of the remaining symbols, according to Figure 13 in the XH article. And calculate the ratios for these symbols. Then duplicate all the single and double-shaft rows to become the triple and X-shaft rows, and calculate their ratios.

You could also add a column for the systematic comma names we use in Sagittal.

These boundaries were carefully chosen so that commas can be named systematically using only their prime factors greater than 3. e.g. When all factors of 2 and 3 are removed from 32768:32805 it leaves only "5", which tells you what it is useful for notating, and its size of 1.95 cents places it in the "schisma" category, therefore it is simply "the 5-schisma".

I should be able to do the image conversions, using MS Paint for aligning and mirroring, and using Gimp to add transparency. I'll try to figure out the proper names, but if I fail I can still name the files something like "3-shaft-7-down.png" for the down-version of the 7th accidental from the left in the row of 3-shaft accidentals.

That would be OK. Thanks for this. I have just finished preparing the file that will tell the forum the mapping from smilie code to file name and description. Although I still need to add the pixel widths of all the new symbols, which will be easier to do after I get the image files from you.

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Fri Nov 04, 2016 6:38 am

by Xen-Gedankenwelt

Dave Keenan wrote:I think your asterisk note above should say "* ambiguous, usually refers to a different comma" rather than "... different accidental".

Thanks, fixed. It's probably debatable whether some of the larger ratios should be called a "comma", so I used "ratio" instead.

Dave Keenan wrote:I note that the commas listed in the wiki tend to be chosen because they vanish in some temperament, and they tend to be named after the temperament in which they vanish. Of course, when looking for commas for use in MOS notations, we need commas that do not vanish. The properties that make commas useful for notation are in some senses the opposite of those that make them useful for creating temperaments. And we would like the names of the commas to convey information about what they notate, rather than where they vanish.

An MOS with a small chroma can be desirable if large chroma stacks are to be expected. For example, compare a Garibaldi[7] notation to a Garibaldi[12] notation. If we use the former to notate all 53-edo pitches, we get notes like F, an F sharpened by an interval slightly smaller than a perfect fourth, and sharper than G and A. The Garibaldi[12] notation, on the other hand, has a small chroma that is a single step in 53-edo, and all 53-edo pitches can be notated without altered nominals switching places.

Of course it's also often desirable that scale steps are distinct, in which case an MOS with a small chroma would be undesirable.

Small comma accidentals can also be useful when defining the nominals. For example, I mentioned above a 6-note notation for the linear 2.5.7 temperament that tempers out the hemimean comma 3136:3125. The generator is 28/25, so a 225:224 accidental (which is equivalent to 126:125 in this temperament) is helpful for the nominals.

Small comma accidentals may also occur when experimenting with ways to use a 7-note staff for a notation of an MOS with more than 7 notes.

Other examples where small accidentals can be useful are temperaments with a lot of periods per octave, and a small generator, like enneadecal, mystery, or birds temperament. And even if the generator is not small, but the difference between one or two periods and a few generators is very small, it'd be useful to use an additional accidental for that.

All in all, I don't think I have strong arguments why small comma accidentals would occur more frequently in MOS notations than other accidentals of similar complexity, and only a few of those arguments probably apply to extremely small commas like 4375:4374.
Notations for rank-3 or higher temperaments (including trivial temperaments that temper out no comma, i.e. just intonation) are a different story, though, where very small chromas should occur more often. I had those notations in mind too, when I created the list, but I guess we should mainly focus on MOS notations for now.

Dave Keenan wrote:I suggest a slightly different approach to this table, which would also eliminate any concerns about ambiguous symbols. You could simply list all 7-limit commas that have an exact representation in Olympian Sagittal. Begin with the list of Olympian ratios for single-shaft symbols in either sag_ji4.par or George's Sagittal-JI.xls spreadsheet, and delete all the rows that are not 7-limit. Then add rows for all the double-shaft symbols which are apotome-complements of the remaining symbols, according to Figure 13 in the XH article. And calculate the ratios for these symbols. Then duplicate all the single and double-shaft rows to become the triple and X-shaft rows, and calculate their ratios.

You could also add a column for the systematic comma names we use in Sagittal.

I think I'll split the list into one for ratios with an exact Olympian representation, and another for ratios w/o. The latter would contain exact multi-accidental representations, or an ambigous single-accidental representation.

Listing the systematic comma names sounds like a good idea.

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Sat Nov 05, 2016 6:04 pm

by Dave Keenan

Xen, Thank you so much for doing that tedious job. I've finally got the rest of the information together and put them up on the forum, tested them and hopefully fixed all my naming and encoding errors. Your work was flawless.

So we finally have the full set of Sagittal symbols available on the forum in all their frightening complexity. View more smilies. Seriously, 90% of everything anyone will ever want to notate can be done with the first 8 pairs (the single-shaft Spartans). You only need the rest if you want to exactly notate more than 2400 different ratios within an octave (half cent resolution).

If you go back to your earlier posts and make an insignificant edit, such as adding a space to the end of a line, and click "Submit", it will force the smiley codes to be re-evaluated and display the new symbols.

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Sun Nov 06, 2016 12:44 am

by Dave Keenan

I see you've been updating the list. Good work.

Yes. I badly overstated the difference between commatic and chromatic commas. Most of the same ratios turn up in both applications. The only difference is in their popularity or usefulness rankings in each application.

I note that one of the criteria for a good choice of accidental for a MOS chroma is that its tempered size should not be too different from its untempered size.

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Sun Nov 13, 2016 12:06 pm

by Xen-Gedankenwelt

Sorry for the late reply! I reread sagittal.pdf and your replies to make sure I understand enough about Sagittal notation with Olympian precision. I think I'm now more or less equal to the task of completing the list of 7-limit accidentals with Olympian precision.

But I have a question:
I know there are some abbreviations for comma size classes, i.e. n, s, k, C, S, M, L for schismina, schisma, kleisma and so on. For example, 5C means 5-comma.
Are there also official abbreviations for the small-semitone, limma and large-semitone class, and the apotome alterations?

Re: List of 7-prime limit / 6125-odd limit accidentals

Posted: Sun Nov 13, 2016 1:46 pm

by Dave Keenan

Xen-Gedankenwelt wrote:Sorry for the late reply! I reread sagittal.pdf and your replies to make sure I understand enough about Sagittal notation with Olympian precision. I think I'm now more or less equal to the task of completing the list of 7-limit accidentals with Olympian precision.

Well done.

But I have a question:
I know there are some abbreviations for comma size classes, i.e. n, s, k, C, S, M, L for schismina, schisma, kleisma and so on. For example, 5C means 5-comma.
Are there also official abbreviations for the small-semitone, limma and large-semitone class, and the apotome alterations?

I'm not sure how official they are, but I use SS MS LS A. I've now added them to my list of categories and boundaries above.

Seriously: Thanks for spotting those errors. Well done. I've now fixed them above.

I note that the boundary between the schisma and kleisma categories is the only one that is somewhat arbitrary, and mostly historical. You'll notice it has very large exponents and isn't mirrored about the half-apotome like the others. For the purpose of distinguishing small ratios with the same primes-above-3, a single schisma/kleisma category would have been sufficient, but in addition to the historical distinction between schismas and kleismas (or semicommas), we thought that it was not good to have a category with a ratio of more than 3:1 between its upper and lower bounds, and a single schisma/kleisma category would have been 6.5:1.

Of course, no finite set of categories is entirely "sufficient" to prevent name clashes between small ratios having the same primes-above-3. It is always possible to alter a ratio by some 3-limit ratio such that it remains in the same size category, but it is rare that the ratio with the higher absolute value of 3-exponent is of any interest. When it is, we simply prepend the word "complex" (symbol "c") to the systematic name. For example the
Pythagorean comma = 3-comma = 3C = [19 -12> ~= 23.5 c, and the
complex Pythagorean comma = complex-3-comma = c3C = [65 -41> ~= 19.8 c